Model predictive control (MPC) is a well-known and widely used control algorithm. The problem of real-time MPC implementation for complex systems is of particular practical interest due to the complexity of the associated optimization problem which is generally intractable in real time. The paper presented deals with this issue making use of the famous dynamical programming idea and reducing the dimensionality of the original optimization problem. The outline of the paper is as follows. The MPC problem is considered for a nonlinear discrete-time system with state and control constraint sets and a quadratic cost functional. The assumptions worth noting are, firstly, the Lipschitz continuity of the right hand side of the system and, secondly, continuity in some sense of the admissible control set with respect to the current state of the system. Employing these properties we are able to prove the Lipschitz continuity of the optimal cost value as a function of the initial state of the system. This result provides us with the opportunity to approximate the minimal value of the last several summands of the cost functional as a function of the intermediate system state by means of precalculating it for a set of state values before the controller is launched. The summands mentioned may be then excluded from the optimization reducing the dimensionality of the problem. The results are followed by a discussion of their limitations and an example of application. It is shown that the simpler the resulting problem, the less smooth it becomes, thus making it necessary to use more data points for the approximation. Another observation is that the smoothness of the problem decreasing far from the set point. The theorems proven in the paper give the reasoning behind these facts but the means of dealing with them are due to further research.